#8873813

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,7)(3,4,8,6)(9,10,15,12)(11,13,17,18)(14,20,23,16)(19,24,22,21)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,18)(14,22)(15,17)(16,24)(19,23)(20,21)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,3)(2,4)(5,8)(6,7)(10,12)(11,17)(16,20)(19,22)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,12,15,10)(11,18,17,13)(14,20,23,16)(19,24,22,21);(9,13,15,18)(10,11,12,17)(14,21,23,24)(16,19,20,22)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,14)(10,16)(11,19)(12,20)(13,21)(15,23)(17,22)(18,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8781009);